2.3.5.11 subgroup: Difference between revisions

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=== Rank-2 temperaments ===

[[~~Schismic~~]] provides ~~two reasonable aproximations to~~ the 2.3.5.11 subgroup, ~~one by finding 11/8 at the triple-augmented second (+23 fifths) through~~ the ~~[[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (~~-~~30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction~~.

Since [[7edo]] provides a relatively accurate approximation of the 2.3.5.11 subgroup for its size, temperaments in this subgroup tend to work well with the 7-form.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at ~~-17~~ gens and 11/8 at ~~-15~~ gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

[[Porcupine]] (15 & 22) provides a simple yet high-damage approximation to the subgroup. It is generated by a submajor second of around 163 cents, representing [[10/9]], [[11/10]], and [[12/11]], tempering out [[55/54]], [[100/99]], and [[121/120]]. Two of these reach [[6/5]]~[[11/9]], and three of them reach the perfect fourth [[4/3]]. It finds 11/8 at –4 generators and 5/4 at –5 generators. The interval at –7 generators represents several important ratios, such as [[25/24]], [[33/32]], [[45/44]], and [[81/80]].

[[Mohaha]] (24 & 31) can be considered an extension to [[meantone]] that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], tempering out [[243/242]]. Here [[11/8]]~[[15/11]] is found as a semi-augmented fourth, and [[11/10]]~[[12/11]] is found as a neutral second, meaning [[121/120]] is also tempered out. The neutral third is very close to 11/9 as a result of the flat fifth, though this means the 11th harmonic is tuned considerably flat.

[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] {{nowrap| (41 & 53) }} mapping, tempering out [[2200/2187]], and another by finding 11/8 at the quadruple-diminished seventh (–30 fifths) through the [[helenus]] mapping, tempering out [[8019/8000]]. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction, and the optimal tuning for prime 11 is closer to the optimal tuning for 5.

[[Gravity]] (7 & 58) also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at –6 gens, 5/4 at –17 gens and 11/8 at –15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

=== Rank-3 temperaments ===

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 === Rank-2 temperaments ===
-[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.
+Since [[7edo]] provides a relatively accurate approximation of the 2.3.5.11 subgroup for its size, temperaments in this subgroup tend to work well with the 7-form.
-[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
+[[Porcupine]] (15 & 22) provides a simple yet high-damage approximation to the subgroup. It is generated by a submajor second of around 163 cents, representing [[10/9]], [[11/10]], and [[12/11]], tempering out [[55/54]], [[100/99]], and [[121/120]]. Two of these reach [[6/5]]~[[11/9]], and three of them reach the perfect fourth [[4/3]]. It finds 11/8 at –4 generators and 5/4 at –5 generators. The interval at –7 generators represents several important ratios, such as [[25/24]], [[33/32]], [[45/44]], and [[81/80]].
+[[Mohaha]] (24 & 31) can be considered an extension to [[meantone]] that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], tempering out [[243/242]]. Here [[11/8]]~[[15/11]] is found as a semi-augmented fourth, and [[11/10]]~[[12/11]] is found as a neutral second, meaning [[121/120]] is also tempered out. The neutral third is very close to 11/9 as a result of the flat fifth, though this means the 11th harmonic is tuned considerably flat.
+[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] {{nowrap| (41 & 53) }} mapping, tempering out [[2200/2187]], and another by finding 11/8 at the quadruple-diminished seventh (–30 fifths) through the [[helenus]] mapping, tempering out [[8019/8000]]. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction, and the optimal tuning for prime 11 is closer to the optimal tuning for 5.
+[[Gravity]] (7 & 58) also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at –6 gens, 5/4 at –17 gens and 11/8 at –15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
 === Rank-3 temperaments ===